We extend Maxwell's representation of harmonic polynomials to $h$-harmonics
associated to a reflection invariant weight function $h_k$. Let $\CD_i$,
$1\le i \le d$, be Dunkl's operators associated with a reflection group.
For any homogeneous polynomial $P$ of degree $n$, we prove the
polynomial $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ is
a $h$-harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and
$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for
$h$-harmonics. We also discuss self-adjoint operators acting on the
space of $h$-harmonics.